请输入您要查询的字词:

 

单词 NormalFamily
释义

normal family


Definition.

A set (sometimes called a family) of continuous functionsMathworldPlanetmath f:XY for some (complete (http://planetmath.org/Complete))metric spaces X and Yis called if each sequence of functions in contains a subsequence which converges uniformly on compact subsets of X to acontinuous function from X to Y.

This definition is often used in complex analysis for spaces of holomorphic functions. It turns out that a sequence of holomorphic functionsMathworldPlanetmath that converges uniformly on compact sets converges to a holomorphic function. So you can replaceX with a region in the complex plane, Y with the complex plane itselfand every instance of “continuous” with “holomorphic” and you get a version of the definition most used in complex analysis.

Another space where this is often used is the space of meromorphic functions.This is similar to the holomorphic case, but instead of using the standardmetric for convergence we must use the spherical metric. That is if σis the spherical metric, then want fn(z)f(z) uniformly on compactsubsets to mean that σ(fn(z),f(z)) goes to 0 uniformly on compactsubsets.

Note that this is a classical definition that, while very often used, is not really with modern naming. In more modern ,one would give a metricon the space of continuous (holomorphic) functions that corresponds toconvergence on compact subsets and then you’d say “precompact set of functions” in such a metric space instead of saying “ family of continuous (holomorphic) functions”. This added generality howevermakes it more cumbersome to use since one would need to define the metric mentioned above.

References

  • 1 John B. Conway..Springer-Verlag, New York, New York, 1978.
随便看

 

数学辞典收录了18232条数学词条,基本涵盖了常用数学知识及数学英语单词词组的翻译及用法,是数学学习的有利工具。

 

Copyright © 2000-2023 Newdu.com.com All Rights Reserved
更新时间:2025/5/4 21:35:09